Topological Galois theory, algebraic geometry and convex geometry
Speaker: Askold Khovanskii, Department of Mathematics, University of Toronto
This series of three lectures is dedicated to three different research areas. In each of them, relations and interlays between seemingly unrelated branches of mathematics are considered. Thus, in the first lecture we discuss a relation between Galois theory and topology, as well as Liouville’s theory of solvability of equations in finite terms. The second lecture is dedicated to connections between algebraic geometry and the geometry of convex polyhedra, particularly their combinatorics and volumes theory. And in the third lecture, we focus on an interplay between the intersection theory of divisors and geometric inequalities, which significantly generalize the classical isoperimetric inequality. The lectures are independent of each other and do not assume any specific mathematical knowledge.
Lecture 1, March 18, 4:10-5 pm, Banatao Auditorium, Sutardja Dai Hall
Solvability of equations in finite terms and topological Galois theory
Abstract: Results on the unsolvability of algebraic and differential equations belong to three very different areas of mathematics. Liouville’s theory explains why the integral of an elementary function usually is not an elementary function and why many differential equations cannot be solved in quadratures. Galois theory provides criteria for solvability of algebraic and linear differential equations in radicals and quadratures, respectively. Topological Galois theory studies topological obstructions to the representability of functions in finite terms. In this lecture, we discuss results on the unsolvability of equations in finite terms based on these theories and discuss some open problems.
Lecture 2, March 19, 4:10-5 pm, Banatao Auditorium, Sutardja Dai Hall
Newton polyhedra and vector-valued Laurent polynomials
Abstract: Newton polyhedra provide a geometric generalization of the degree of polynomials and connect algebraic geometry to the geometry of convex polyhedra. This connection is useful in both directions. On the one hand, explicit answers are given to problems of algebra in terms of the geometry of polyhedra. On the other hand, algebraic theorems of general character give significant information about the geometry of polyhedra. Recently, an unexpected wide generalization of Newton polyhedra theory for vector-valued Laurent polynomials was discovered. It is related to the beautiful results of June Huh, for which he was awarded the Fields Medal in 2022. In this lecture, we will discuss these results and connections.
Lecture 3, March 20, 4:10-5 pm, 60 Evans Hall (part of Colloquium series)
Newton-Okounkov bodies and isoperimetric type inequalities in algebraic geometry
Abstract: The theory of Newton–Okounkov bodies is based on the study of sub-semigroups in the lattice of integral points. It provides direct relations and analogies between the theory of convex bodies and algebraic geometry. There are many classical inequalities between mixed volumes of convex bodies that generalize the famous isoperimetric inequality, which was known already to the ancient Greeks. Each of these inequalities corresponds to a similar inequality between intersection indices of divisors in algebraic geometry. In this lecture, we will discuss these geometric and algebraic inequalities and will derive geometric inequalities from algebraic ones. For an overview of this area, see https://arxiv.org/abs/2502.13099.
Biography of Speaker
Askold Khovanskii received his PhD in mathematics from the Steklov Mathematical Institute in 1973 and received his Doctor of Sciences there in 1988. After completing his PhD, he worked as a research fellow at the Institute for System Studies of the Russian Academy of Sciences. Since 1995, he has been a professor of mathematics at the University of Toronto. He is known for developing the topological Galois theory, a new branch of mathematics that connects classical Galois theory with topology. He also introduced the theory of fewnomials, leading to the solution of several longstanding mathematical problems and influencing the development of O-minimal structure theory. In addition, he is one of the creators of the theories of Newton polyhedra and Newton–Okounkov bodies. For his outstanding contributions to mathematical research, Khovanskii was awarded the Jeffery–Williams Prize and elected a Fellow of the Royal Society of Canada.